Scalar spectral index in potential slow-roll parameters
= Scalar spectral index in potential slow-roll parameters
{title2=$n_s\simeq1-6\epsilon_V+2\eta_V$}
The leading scalar spectrum is proportional to $V^3/V'^2$. Along a slow-roll trajectory, $d\phi/d\ln a\simeq-M_{\rm pl}^2V'/V$ and $d\ln k/d\ln a=1-\epsilon_H$. Differentiating at horizon exit and keeping first order gives $n_s-1=-6\epsilon_V+2\eta_V$. The derivative-factor correction contributes at second order.